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Square roots

Solving equations of the form

The theorem of Pythagoras

The converse of the Pythagorean theorem

Pythagorean triples

Problem solving using Pythagoras

Three dimensional problems (Extension)

x k2

Contents:

1Pythagoras

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Y:\HAESE\SA_09-6ed\SA09-6_01\011SA09-6_01.CDR Wednesday, 18 October 2006 2:11:40 PM PETERDELL

OPENING PROBLEM

HISTORICAL NOTE

12 PYTHAGORAS (Chapter 1)

Over 3000 years ago the Egyptians were familiar withthe fact that a triangle with sides in the ratio 3 : 4 : 5 wasa right angled triangle. They used a loop of rope with

12 knots equally spaced along it to make right angledcorners in their building construction.

Around 500 BC, the Greek mathematician Pythagorasfound a rule which connects the lengths of the sides

of any right angled triangle. It is thought that he dis-

covered the rule while studying tessellations of tiles on

bathroom floors.

Such patterns, like the one illustrated were common on

the walls and floors of bathrooms in ancient Greece.

The discovery of Pythagoras Theorem led to the discovery of a different type of number

which does not have a terminating or recurring decimal value and yet has a distinct place

on the number line. These sorts of numbers likep

13 are called surds and are irrationalnumbers.

Consider the following questions:

1 If the fire truck parks so that the base of the ladder

is 7 m from the building, will it reach the third floorwindow which is 26 m above the ground?

2 The second floor window is 22 m above the ground.By how much should the length of the ladder be re-

duced so that it just reaches the second floor win-

dow?

3

The study of Pythagoras Theorem and surds will enable you to solve questions such as

these.

corner

line of one side of building

make rope taut

take hold of knots at arrows

26 m

22 m

2 m

25 m

7 m

A building is on fire and the fire department are called. They need to

evacuate people from the third floor and second floor windows with the

extension ladder attached to the back of the truck. The base of the ladder is

m above the ground. The fully extended ladder is m in length.2 25

For many centuries people have found it

necessary to be able to form right angled

corners. Whether this was for constructing

buildings or dividing land into rectangular

fields, the problem was overcome by rela-

tively simple means.

If the ladder gets stuck in the fully extended position, how much further should the

fire truck move from the building to just reach the second floor window?away

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Y:\HAESE\SA_09-6ed\SA09-6_01\012SA09-6_01.CDR Thursday, 21 September 2006 4:17:46 PM PETERDELL

PYTHAGORAS (Chapter 1) 13

Recall that 5 5 can be written as 52 (five squared) and 52 = 25.Thus, we say the square root of 25 is 5 and write

p25 = 5.

Note:p

reads: the square root of .

Finding the square root of a number is the opposite of squaring a number.

To find the square root of 9, i.e.,p9, we need to find a positive number which when

multiplied by itself equals 9.

As 3 3 = 9, the number is 3, i.e., p9 = 3:

Notice that: The symbol p is called the square root sign. pa has meaning for a > 0, and is meaningless if a < 0. pa > 0 (i.e., is positive or zero).

1 Evaluate, giving a reason:

ap16 b

p36 c

p64 d

p144

ep81 f

p121 g

p4 h

p1

ip0 j

q1

9k

q1

49l

q25

64

Finding the square root of numbers which are square numbers such as 1, 4, 9, 25, 36, 49etc. is relatively easy. What about something like

p6?

What number multiplied by itself gives 6? If we use a calculator ( 6 ) we get2:449 489 743.

Here we have a non-recurring decimal which is only an approximation ofp6.

p6 is an

example of an irrational number, as it cannot be written as a fraction of any two integers.

In this chapter we will deal with irrational numbers of square root form called surds.

IRRATIONAL NUMBERS

EXERCISE 1A

SQUARE ROOTSA

Evaluate, giving a reason: ap49 b

q1

16

a As 7 7 = 49 b As 14 1

4= 1

16

thenp49 = 7 then

q1

16= 1

4

Self TutorExample 1

=

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Y:\HAESE\SA_09-6ed\SA09-6_01\013SA09-6_01.CDR Wednesday, 23 August 2006 10:54:18 AM DAVID3

14 PYTHAGORAS (Chapter 1)

Between which two consecutive integers doesp11 lie?

Sincep9 = 3 and

p16 = 4 and 9 < 11 < 16

thenp9 0.

Self TutorExample 12

EXERCISE 1E

PYTHAGOREAN TRIPLESE

5

4

3

Let be

the largestnumber!

c

a

bc

is not a Pythagorean triple as these numbers are not all integers.

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Y:\HAESE\SA_09-6ed\SA09-6_01\023SA09-6_01.CDR Friday, 15 September 2006 11:01:40 AM PETERDELL

INVESTIGATION 4 PYTHAGOREAN TRIPLES SPREADSHEET

24 PYTHAGORAS (Chapter 1)

The result in question 3 enables us to write down many Pythagorean triples using multiples.

It also enables us to find the lengths of sides for triangles where the sides may not be integers.

4 Use multiples to write down the value

of the unknown for the following triangles.

Given:

a b c d

5 Use multiples to write down the value of

the unknown for the following triangles.

Given:

a b c d

The best known Pythagorean triple is f3, 4, 5g.Other triples are f5, 12, 13g, f7, 24, 25g and f8, 15, 17g.

There are several formulae that will generate Pythagorean triples.

For example, 3n, 4n, 5n where n is a positive integer. A spreadsheet will quicklygenerate sets of triples using these formulae.

SPREADSHEET

5

4

3

x

8

6

25

x

15

30

x

18

x

2

1.5

Use multiples to write down the

value of the unknown for the

following triangles.

Given:

a b

a 5 : 12 : 13 = 10 : x : 26

) x = 12 2i.e., x = 24

b 5 : 12 : 13 = 50 : 120 : y

) y = 13 10i.e., y = 130

Self TutorExample 13

12

135

x

2610

y 50

120

2 10

2 10

1

p2

2

x

3x x

p210

x

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Y:\HAESE\SA_09-6ed\SA09-6_01\024SA09-6_01.CDR Thursday, 24 August 2006 2:00:48 PM DAVID3

PYTHAGORAS (Chapter 1) 25

Right angled triangles occur in many realistic problem solving situations. When this happens,

the relationships involved in Pythagoras Theorem can be used as an aid in solving the

problem.

The problem solving approach usually involves the following steps:

Step 1: Draw a neat, clear diagram of the situation.

Step 2: Mark known lengths and right angles on the diagram.

Step 3: Use a symbol, such as x, to represent the unknown length.

Step 4: Write down Pythagoras Theorem for the given information.

Step 5: Solve the equation.

Step 6: Write your answer in sentence form (where necessary).

PROBLEM SOLVING USING PYTHAGORASF

filldown

1 Open a new spreadsheet and

enter the following:

2 Highlight the formulae in B2, C2 and D2, andfill these down to row 3. Now highlight all theformulae in row 3 and fill down to row 11.

This should generate 10 sets of Pythagoreantriples that are multiples of 3, 4 and 5.

3 Change the headings in B1 to 5n, C1 to 12n and D1 to 13n.

Now change the formulae in B2 to =A2*5 in C2 to =A2*12 in D2 to =A2*13:

Refill these to row 11. This should generate 10 more Pythagorean triples that are mul-tiples of 5, 12 and 13.

4 Manipulate your spreadsheet to find other sets of Pythagorean triples based on:

a f7, 24, 25g b f8, 15, 17gExtension:

5 The formulae 2n+1, 2n2+2n, 2n2+2n+1 will generate Pythagorean triples.Devise formulae to enter in row 2 so your spreadsheet calculates the triples.

6 How could you check that the triples generated above are Pythagorean triples using

your spreadsheet? Devise suitable formulae to do this.

7 Use a graphics calculator to generate sets of Pythagorean triples in a TABLE OR

LIST.

What to do:

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